apply
Feynman diagram methods as a starting point to understand reactions

Identify
symmetry constraints

recognize
important processes and their implications

To start lets take a simple world that has a charged
particle and a E&Mfield. As in all
particle physics our formulation needs to incorporate realativity.

If we imagine that the particle is an electron we will need
to specify some parameters that identify the particle. Above we find charge,
mass and spin as particle labels.In
addition we will also need to specify other aspects of the state that
characterize the particles motion.This
could be characterized by providing for example the particle momentum.

In relativity the representation of position and momentum is
as 4-vectors.

[Wikipedia: “In terms of covariance and contravariance
of vectors, lower indices represent (components of!) covariant
vectors (covectors),
while upper indices represent (components of!) contravariant
vectors (vectors): they transform covariantly (resp.,
contravariantly) with respect to change of coordinates.”Subtle pointèThe autohor wants to highlight the fact that you
can define the two vectors as components time basis vectors. He prefers to keep
the idea of a basis vector separated. Thus you define one basis and two types
of vectors.Only the components of these
vectors are presented as values.]

Wolfram

Contravariant

There are Lorentz invariants

E&M

Here the electric and magnetic fields are grouped to form a
second rank tensor. However the field can also be represented in terms of
fields

which tells us that there are no
"true" magnetic charges, and the magnetodynamics equation

which tells us the change of the
magnetic field with respect to time plus the curl of the Electric field is equal to zero (or,
alternatively, the curl of the electric field is equal to the negative change
of the magnetic field with respect to time). With the electromagnetic tensor,
the equations for magnetism reduce to

Decide that the ability to provide the first order Feynman
will be important. Therefore the students should be able to draw the diagrams
for any interaction. The focus will be primarily on the quark and lepton level
but strong interactions will be mediated on the long-range scales by quark
exchange. Simplest model for proton neutron interactions is pion exchange.

*******************************

Following excerpted from Stanford website

http://www2.slac.stanford.edu/vvc/theory/feynman.html

·Left-to-right in the
diagram represents time; a process begins on the left and ends on the right.

·Every line in the
diagram represents a particle; the three types of particles in the simplest
theory (QED) are:

NOTE: (There are different conventions for the direction of
time. Some choose to have time develop vertically and so the diagrams are
rotated.

A photon produces an electron and a positron (an electron-positron
pair)

An electron and a positron meet and annihilate
(disappear), producing a photon

Need to identify internal lines and external lines when
diagrams are assembled to describe processes.Lines that ultimately enter or leave are real particles and must have a
physical mass corresponding to the particle type.Internal lines are mediating the interaction.
That is they are exchanged in order to have a momentum and energy transfer or
interaction between two physical particles.These field particles do not have to have a correct mass based on the
particle type.At all vertices the
momentum and energy are conserved. Sum of all the incoming is equal to the
outgoing for the three (four) particles that combine to from a vertex.

So the QED vertex

One can rotate any arm as follows:

We will need to examine which rotations are
significant.In order to do this we need
to identify internal and external lines. Therefore we draw the lowest order
diagram that represents a process:

A+BèC+D

Rotating external lines is only significant if the line move
across the 90o line. This causes the particle to turn into its
antiparticle and the reaction changes.

Aè antiB + C+ D
as shown above.

Moving particle from one side to the other or rotating
external arms is based on crossing
symmetry.If one is actually
performing a calculation the question remains do elements of the underlying calculation
change when one explores the crossing symmetry related diagrams.

change
the sign of the four momentum so that the particle is now a negative
energy state: p is the four momnetum

A(p1)+B(p2)èC(p3)+D(p4)same amplitude A(p1) èC(p3)+D(p4) +antiB(-p2)

fermions
have an additional factor of -1 for the amplitude

Rotating internal lines simply changes the time ordering of
the vertices.All such time orderings
are assumed to be included so only one of the diagrams is drawn.

Therefore there is only the need to draw on of these
drawings to describe the process A+BèC+D

However there may be other drawings. These can be found by
deciding which of the external lines are meeting at a vertex.

If we look at the possibilities and calculate the 4-momentum
given to the internal linethen we have:

A B vertex

PAμ + PBμ

s channel

A C vertex

PAμ - PCμ

t channel

A D vertex

PAμ - PDμ

u channel

The allowable vertices depend on the types of particles.

e+ + e-è e+ + e-(E&M)

Allows two of the diagrams above, the incoming particles
(A,B) can form a vertex or outgoing (A,C) can form a vertex but not (A,D).

*******************************z

Weak vertices

uèd,cès,νèe-
,e+è
νbar

These emission graphs are W+

dèu,sèc,e-è
ν,νbarè
e+

These emission graphs are W-

uèu,dèd,νè
ν,e+è
e+

These emission graphs are Zo

uèd,cès,νèe-
,e+è
νbar

These absorption graphs are W-

dèu,sèc,e-è
ν,νbarè
e+

These absorption graphs are W+

uèu,dèd,νè
ν,e+è
e+

These absorption graphs are Zo

èubard,ècbars,è
e- νbar

These production graphs are W-

èdbaru,è
sbarc,èe+
ν

These production graphs are W+

ubardè,cbarsè,e-
νbarè

These annihilation graphs are W-

dbaruè,sbarcè,e+ νè

These annihilation graphs are W+

èubaru,ècbarc,è
ν νbar,è
e- e+

These annihilation graphs are Zo

ubaruè,cbarcè,ν
νbarè,e- e+è

These production graphs are Zo

Due to quark mixing the any d,s,b can be replaced by a
d,s,b.

Here are some weak processes from the lepton sector.The external lines show the fundamental
fields to be the (e,νe) and (μ,ν μ).For the weak interaction there is no
difference between electron and neutrino. They are manifestations of the same
particle. The interactions will need W+, W- ,Zo
to mediate the interaction so they will be included.

(ASIDE: Also to carefully define the states the electron and
muon need to have right and left handed versions.eR, eL, μR,μL. These are spin projection that are chosen by
projecting onto the momentum direction. Only left handed leptons are present in
the above diagrams. This detail can be neglected without difficulty when
establishing and overview of the interactions.)

Most of the hadrons will interact over long distances by
pion exchange because color singlets are not allowed.Typically we can follow these by showing the
quark lines. The gluons have not been shown in the following diagrams.

Very close
range interactionStandard
nuclear force

P+NèP+NP+NèP+N

Diagrams on the left are not color singlets so they must be
very short range. While the diagrams on the right exchange a particle that is a
color singlet.

Very close
range interactionStandard
nuclear force

P+NèP+NP+NèP+N

###########################

Weak interaction

fundamental
vertexspectator
quarks added

######################################x

Weak

fundamental vertexspectator quarks
added

The QCD interactions are complicated because the quarks come
in three colors but the color structure of objects that make up composite
systems is not typically shown. In principle one can simply add gluons in the
same manner that one adds photons. The only change will be in the color which
is not shown.The usefulness of the
Feynman diagrams requires a bit more care. Since there can be appreciable gluon
exchange an expansion in terms of graphs is only appropriate in some
situations.